How do I construct this japanese bowl?
You could try it this way:


Put your picture as a Background image.





Create a plane, rotate it on the X axis to have it in front view, apply the rotation.
In Front Ortho view, use the picture as a model to build your mesh, mirror the mesh to make it easier.





Switch to Right Ortho view and continue to model the mesh. You'll have to shift the mesh from its origin.





Come back to Object mode. Create an empty at the same point as the object origin.





Give your object an Array modifier, give it a Count of 8, deactivate Relative Offset, activate Object Offset, choose the empty as object.





Rotate the empty 45° on the Z axis.





In Edit mode, give some corrections to make the vertices stick approximately with their arrayed copies.





Duplicate the empty + object to keep a copy somewhere on another layer.
Apply the modifiers.
Go in Edit mode, remove doubles.





Dissolve the useless vertical edges.





Make some corrections.





Now you have your basic bowl. Of course you'll probably want it to be a bit more sophisticated, like adding some bevels, etc... in that case you can keep just 1/8 of the bowl and again spin it around the empy with an array, add your bevels on the original mesh, keep only quads, etc...